Extend braid-group computation of fundamental variables beyond finite type

Prove that for any generalized Cartan matrix (not necessarily of finite type), the fundamental variables of the cluster algebras associated with double Bott-Samelson cells (i.e., the cluster algebras arising from signed words) can be computed via the braid group action, generalizing the finite-type result that expresses these variables through the action of braid group automorphisms on the corresponding Serre generators.

Background

The authors establish that, in finite type, the fundamental variables in cluster algebras associated with double Bott-Samelson cells can be obtained via a braid group action on the relevant quantum (virtual) Grothendieck rings. This provides a powerful computational mechanism and connects cluster structures to representation-theoretic symmetries.

They conjecture that the same mechanism should extend to arbitrary generalized Cartan matrices, suggesting the use of braid symmetries developed for general types. Proving this would substantially broaden the applicability of the braid-action framework and further link cluster algebra structures to deep properties of quantum groups across all types.